Apparatus and Method for Transforming a Coordinate System to Simulate an Anisotropic Medium

ABSTRACT

A method and apparatus for simulating a mesh element of an anisotropic medium are provided. A unitary transformation is applied to an initial coordinate system of the mesh element by a transformation module to produce a transformed reference coordinate system of the mesh element. Maxwell&#39;s equations for the mesh element are solved by an update generation module using computational methods to obtain an electric field tensor and an electric displacement field tensor within the mesh element. A unitary transformation to the electric field tensor and the electric displacement tensor are performed by a transformation module to calculate a corresponding electric field tensor and electric displacement tensor for the mesh element in the initial coordinate system.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from U.S. Provisional Application No.61/650,774, filed on May 23, 2012, the entire contents of which areincorporated herein by reference.

TECHNICAL FIELD

The following relates generally to the simulation of an anisotropicmedium.

BACKGROUND

It is desirable to solve Maxwell's equations for various electric andmagnetic media using the finite-difference time-domain (FDTD) method tosimulate electromagnetic properties of the media. Such simulations canbe used to determine the FDTD method involves spatially discretizing thevolume of the medium being simulated to form a mesh of individualelements. The individual elements can be referred to as mesh cells orYee cells. The FDTD method is routinely used to simulate isotropicmaterials.

It can also be desirable to also solve equations involving anisotropicmedia to model the electrical and magnetic properties of media invarious systems and devices. For example, it may be desirable tosimulate the electromagnetic properties of liquid crystals in theconception, design, and testing of liquid crystal displays (LCDs) andelectrical drivers. However, anisotropy complicates the solution ofMaxwell's equations using the FDTD method.

Solving Maxwell's equations in dispersive anisotropic materials enablesthe simulation of electromagnetic properties of device designscomprising such materials. However, simulation of anisotropic materialscan be complicated by spatially varying permittivity. For example,liquid crystal modeling can be challenging in cases where theorientation of liquid crystal molecules is a function ofthree-dimensional space.

The simulation of electromagnetic properties is also complicated incases where all terms in the permittivity tensor are dispersive. Theseterms may be approximated by FDTD models including Plasma-Drude,Lorentz, Debye and more general multi-pole expansion dispersive models.

Although attempts have been made to overcome the above deficiencies,these have focused on sub-element smoothing or conformal mesh algorithmsto reduce the number of errors that occur at interfaces betweendifferent media on the finite sized mesh used in the simulation. SolvingMaxwell's equations in the simulation of electromagnetic properties ofgeneral dispersive, anisotropic, spatially varying media remainsdifficult due to challenges in computing numerical solutions toMaxwell's equations.

It is an object of the present invention to mitigate or obviate at leastone of the above disadvantages.

SUMMARY

In one aspect, a computer-implemented method of simulating predefinedproperties of a mesh element defined by an initial reference frame of ananisotropic medium is provided. The method comprises generating atransformed reference frame of the mesh element by applying a unitarytransformation to the initial reference frame, the unitarytransformation for diagonalizing a permittivity tensor associated withthe medium and utilizing the unitary transformation to generate atransformed flux density tensor and a field tensor in the transformedreference frame based on a flux density tensor and a field tensor of themedium in the initial reference frame. The transformed flux densitytensor and the transformed field tensor for the mesh element are updatedto obtain the predefined properties based on the diagonalizedpermittivity tensor.

In another aspect, a method of transforming a coordinate system of ananisotropic medium is provided. The method comprises applying a unitarytransformation to diagonalize a permittivity matrix attributed to themedium.

In yet another aspect, there is provided an apparatus for simulatingpredefined properties of a mesh element defined by an initial referenceframe of an anisotropic medium. The apparatus comprises a transformationmodule operable to generate a transformed reference frame of the meshelement by applying a unitary transformation to the initial referenceframe, the unitary transformation for diagonalizing a permittivitytensor associated with the medium, and an update generation moduleoperable to utilize the unitary transformation to generate a transformedflux density tensor and a field tensor in the transformed referenceframe based on a flux density tensor and a field tensor of the medium inthe initial reference frame. The update generation module is furtheroperable to update the transformed flux density tensor and thetransformed field tensor for the mesh element to obtain the predefinedproperties based on the diagonalized permittivity tensor.

In yet another aspect, there is provided apparatus for transforming acoordinate system of an anisotropic medium comprising a transformationmodule operable to apply a unitary transformation to diagonalize apermittivity matrix attributed to the medium.

The unitary transformation may be applied to the initial reference frameto align the transformed reference frame with an anisotropic axis of themedium.

The inverse of the unitary transformation may be applied to the updatedfield tensor and flux density tensor to generate a corresponding fieldtensor and flux density tensor for the mesh element in the initialreference frame for subsequent comparison with the mesh element with oneor more other mesh elements in the initial reference frame.

The unitary transformation may be applied to each of a plurality of meshelements of the medium. The initial coordinate system may be a globalcoordinate system. The global coordinate system may be common to each ofthe mesh elements. The transformed reference coordinate system may be alocal coordinate system. The mesh element may be represented by a Yeecell.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments will now be described by way of example only with referenceto the appended drawings wherein:

FIG. 1 is a two dimensional representation of an anisotropic medium;

FIG. 2 is an example block diagram of a system according to the presentinvention;

FIG. 3 is a flow chart illustrating an example method of solving theMaxwell equations; and

FIG. 4 is an example diagram of a Yee cell.

DETAILED DESCRIPTION OF THE DRAWINGS

Embodiments will now be described with reference to the figures. It willbe appreciated that for simplicity and clarity of illustration, whereconsidered appropriate, reference numerals are repeated among thefigures to indicate corresponding or analogous elements. In addition,numerous specific details are set forth in order to provide a thoroughunderstanding of the embodiments described herein. However, it will beunderstood by those of ordinary skill in the art that the embodimentsdescribed herein may be practiced without these specific details. Inother instances, well-known methods, procedures and components have notbeen described in detail so as not to obscure the embodiments describedherein. Also, the description is not to be considered as limiting thescope of the embodiments described herein.

It will also be appreciated that that any module, component, server,computer, terminal or device exemplified herein that executesinstructions may include or otherwise have access to computer readablemedia such as storage media, computer storage media, or data storagedevices (removable and/or non-removable) such as, for example, magneticdisks, optical disks, or tape. Computer storage media may includevolatile and non-volatile, removable and non-removable media implementedin any method or technology for storage of information, such as computerreadable instructions, data structures, program modules, or other data.Examples of computer storage media include RAM, ROM, EEPROM, flashmemory or other memory technology, CD-ROM, digital versatile disks (DVD)or other optical storage, magnetic cassettes, magnetic tape, magneticdisk storage or other magnetic storage devices, or any other mediumwhich may be used to store the desired information and which may beaccessed by an application, module, or both. Any such computer storagemedia may be part of the device or accessible or connectable thereto.Any application or module herein described may be implemented usingcomputer readable/executable instructions that may be stored orotherwise held by such computer readable media.

A system and method for transforming a coordinate system to simulatepredetermined properties of anisotropic media is provided. In oneaspect, the method provides numerical solutions of Maxwell's equationsfor anisotropic dispersive media defined by a global coordinate system.The method provides numerical solutions by transforming a localreference frame of each element to align with the axis of theanisotropic medium being modeled. Numerical solutions arecomputationally less expensive when the reference frame of thecomputation is aligned with the axis of the anisotropic medium, as isfurther described herein.

In another aspect, an apparatus comprising a transformation module isprovided for solving Maxwell's equations for anisotropic media. In someexamples, the anisotropic media is dispersive. The system enables thecomputation of numerical solutions of Maxwell's equations foranisotropic dispersive media in a simulation to obtain predeterminedproperties of the medium. Predetermined properties of the anisotropicmedium include, for example, the electric field, electric displacementfield, magnetic field, and magnetic flux of the medium. The anisotropicand dispersive properties of the media are, in some cases, spatiallydependent. For example, the anisotropic and dispersive properties of themedia may vary discretely or continuously over the media.

The method comprises transforming a local coordinate system of ananisotropic medium to a reference frame in which a matrix representingthe permittivity of the medium is a diagonal matrix. Once thetransformation is complete, Maxwell's equations can be solved usingvarious dispersive models to simulate predetermined properties ofdispersive anisotropic media.

Simulation of electromagnetic properties can be complicated bynon-diagonal terms in the electric and magnetic permittivity tensor ofthe material being studied. For example, non-diagonal terms arise whenthe axis of a polarizing filter in an LCD is not aligned with the axisof the liquid crystal. An example medium is a liquid crystal display(LCD) defined by a global coordinate system and comprising a pluralityof mesh elements that are approximately identical in construction, eachbeing defined by a local coordinate system that is initially alignedwith the global coordinate system. The matrix representing thepermittivity of the medium is diagonal when the coordinate system isaligned with the axis of the anisotropic medium, for example, themolecular axis of a liquid crystal molecule.

A transformation module 150, as shown in FIG. 2, is operable totransform the local coordinate system to diagonalize the matrix byaligning the local coordinate system with the axis of the anisotropicmedium. The local coordinate system is also referred to herein as alocal reference frame.

When the local coordinate system at each mesh element is aligned withthe molecular axis of the liquid crystal, Maxwell's equations can besolved more readily, as non-diagonal terms in the permittivity tensordisappear. In the case of an LCD, the permittivity of each mesh elementis substantially identical, allowing the application of a commondispersive material model for each mesh element in the LCD when solvingMaxwell's equations. Upon solving Maxwell's equations, thetransformation module may optionally transform the local reference frameto align the local reference frame with a global reference frame.

Referring to FIG. 1, an example anisotropic medium 102 is shown. Forillustrative purposes, FIG. 1 is shown in two dimensions, however, itwill be appreciated that the concepts presented herein will be equallyapplicable to three dimensional media.

The medium 102 may, for example, comprise birefringent liquid crystalmolecules oriented along a particular axis. The permittivity of themedium 102 depends on the relative orientation of the electromagneticfield interacting with the medium. The medium 102 can be assigned aglobal reference frame used to describe the properties of the media. Themedia can then be discretized into a plurality of elements and each ofthese discrete elements can be assigned an individual local referenceframe.

The local reference frame may align with the global reference frame, theglobal reference frame being consistent across all elements. A localreference frame aligned with the global reference frame is representedby numeral 106 prior to transformation. As is shown in FIG. 1, the localreference frame is transformed from alignment with the global referenceframe to be in alignment with the axis of the medium 102. Thetransformed reference frame is represented by numeral 104.

Turning now to FIG. 2, a simulation apparatus 140 is provided forsimulating predetermined properties of anisotropic media. The simulationapparatus 140 comprises a transformation module 150, which comprises, oris linked to, a processor 152 and a memory 154. The transformationmodule 150 is also in communication with an update generation module156.

The transformation module 150 is operable to perform coordinatereference frame transformations on one or more mesh elements. Thetransformation module 150 is operable to transform the local referenceframe of an element in a model to align the local reference frame withthe axis of a medium, as is further described below. For example, thetransformation module is operable to perform the coordinatetransformation shown in FIG. 1. The transformation module 150 is alsooperable to transform the local reference frame to be re-aligned with aglobal reference frame.

The update generation module 156 is operable to generate solutions tothe Maxwell equations for one or more mesh elements in a transformedreference frame, as is described in detail below with reference to FIGS.3 and 4.

The transformation module 150 transforms the local reference framethrough a unitary transformation to diagonalize the permittivity tensor.Since the transformation module 150 performs a unitary transformation,any properties of the permittivity tensor matrix that enable numericalstability when solving Maxwell's equations can be retained. The unitarytransformation may be real or complex while retaining properties of thepermittivity tensor.

For example, properties that can be retained under a unitarytransformation include a positive definite matrix will remain positivedefinite and a Hermitian matrix will remain Hermitian. The use of aunitary transformation enables a simulation of a system that is stablysolved by FDTD in its original reference frame to remain stable in thetransformed reference frame.

Upon the transformation module 150 transforming the local coordinatesystem of the matrix, the update generation module 156 employs thefinite difference time domain (FDTD) method to solve Maxwell'sequations. In general, the FDTD method is well known in the art.Briefly, the inverse permittivity tensor converts the electricdisplacement field to the electric field. Similarly, the inversepermeability tensor converts the magnetic flux density to the magneticfield in magnetic field calculations.

Referring to FIG. 3, a process flow of predetermined properties of ananisotropic medium being simulated is provided. The transformationmodule 150 first transforms the local reference frame to align the localreference frame with the axis of the anisotropic medium being simulated,as is represented by 202. This causes the permittivity tensor to bediagonal, thereby simplifying calculations conducted by the updategeneration module 156. In 204, the update generation module 156 solvesthe Maxwell equations to determine the electric displacement field andelectric field based on the diagonalized permittivity tensor. Thesolution of the Maxwell equations will be described in further detailbelow. In 206, the transformation module 150 optionally transforms theelectric displacement field and the electric field back to the globalreference frame to enable further calculations to simulate the medium.

Since electrical properties of a medium affect the magnetic propertiesof a medium and vice versa, one example implementation of the FDTDmethod is to alternate between solving the electric field and themagnetic field of the medium in question to simulate either theelectrical, magnetic, or both, properties of the medium. The electricfield and magnetic field can be solved iteratively and alternatelyaccording to the following equations:

$\frac{\partial{\overset{arrow}{D}(t)}}{\partial t} = {\overset{arrow}{\nabla}{\times {\overset{arrow}{H}(t)}}}$${\overset{arrow}{D}(\omega)} = {{ɛ(\omega)}{\overset{arrow}{E}(\omega)}}$$\frac{\partial{\overset{arrow}{B}(t)}}{\partial t} = {{- \overset{arrow}{\nabla}} \times {\overset{arrow}{E}(t)}}$${\overset{arrow}{B}(\omega)} = {{\mu (\omega)}{\overset{arrow}{H}(\omega)}}$

Where {right arrow over (D)} is the electric displacement field, {rightarrow over (E)} is the electric field, {right arrow over (B)} is themagnetic flux density and {right arrow over (H)} is the magnetic field,e is the electric permittivity tensor μ is the magnetic permeabilitytensor, ω is the angular frequency, and t is time. The electricpermittivity and magnetic permeability tensors must be symmetric andpositive semidefinite or positive definite when using the FDTD method toallow for numerical stability.

The electric permittivity tensor, magnetic permeability tensor, electricfield and magnetic field are complex in the frequency domain but areoften real-valued in the time domain. An exception to this is whensolving the magneto optical effect of an anisotropic dispersivematerial, complex values can be used for the electric permittivitytensor and electromagnetic fields in the time domain. In this case, thepermittivity are Hermitian and positive semidefinite or positivedefinite. When ε and μ are dispersive and complex valued in thefrequency domain, various material models well known in the field areused to solve the equations in the time domain using real valued E and Hfields. These material models may solve for Plasma-Drude, Lorentz, andDebye and more general multi-pole dispersive materials.

As outlined above, the electric field and magnetic field equations canbe iteratively and alternately solved to determine the electromagneticproperties of the medium being simulated. In the example of an LCDcomprising a plurality of identical mesh elements, these equations canbe solved using the same material model, with the same parameters, foreach mesh element of the liquid crystal material upon the transformationmodule 150 transforming the reference frame.

Although the following is described with reference to solving for theelectric field equation, the same principles may be applied to themagnetic field domain. As mentioned above, U is a unitary transformationU⁻¹=U^(†), where U^(†) is the complex conjugate transpose of U. Asmentioned above, the use of a unitary transformation by thetransformation module 150 preserves the stability in the numericalcomputation. The transformation module 150 is operable to transform theelectric field (E), electric displacement field (D), and thepermittivity (ε) according to the following:

{right arrow over (D)}′=U ^(†) {right arrow over (D)}

{right arrow over (E)}′=U ^(†) {right arrow over (E)}

ε′=U ^(†) εU

The relationship between the electric displacement field and theelectric field is preserved in the transformed reference frame accordingto the following:

{right arrow over (D)}=ε{right arrow over (E)}=UU ^(†) εUU ^(†) {rightarrow over (E)}

U ^(†) {right arrow over (D)}=ε′U ^(†) {right arrow over (E)}

D′=ε′E′

The unitary matrix U can be constructed from column eigenvectors of ε,rendering E′ a diagonal matrix (i.e. a square matrix having no termsoutside its diagonal). By definition, a diagonal matrix does not includenon-diagonal terms. A matrix with only diagonal terms can be manipulatedmore easily and with fewer computational steps using FDTD. For example,the manipulations may comprise manipulations that are commonly used formaterials with only a diagonal permittivity or permeability tensor.

For example, the manipulation {right arrow over (E)}=ε⁻¹{right arrowover (D)} may also be written as: {right arrow over(E)}=U(ε′)⁻¹U^(†){right arrow over (D)}. Because the transformation isunitary, (ε′)⁻¹ will be numerically stable, as it will have the sameproperties as ε⁻¹ that ensures numerical stability.

Iterations of the FDTD method can be implemented according to FIG. 3. In202, the transformation module 150 transforms the displacement andelectric fields D′ and E′ to a local coordinate system according to:

{right arrow over (D)}′=U ^(†) {right arrow over (D)}

{right arrow over (E)}′=U ^(†) {right arrow over (E)}

In 204, the update generation module 156 solves the Maxwell equationsfor D′ and E′ using ε′ of the material. It will be appreciated that ε′may comprise a dispersive permittivity or a nonlinear permittivity. Theequation can, for example, be of the form: {right arrow over(E)}′=(ε′)⁻¹{right arrow over (D)}, however this depends on the preciseform may depend on the type of FDTD update. Due to the transformation,the same update normally used for diagonal permittivity updates can beapplied.

In 206, the transformation module 150 transforms the generated tensorsassociated with the electric displacement field and the electric fieldinto the global reference frame. The electric displacement field andelectric field can be solved for the global reference frame according tothe equations:

{right arrow over (D)}=U{right arrow over (D)}′

{right arrow over (E)}=U{right arrow over (E)}′

The remaining curl equations can then be solved without modification.The transformation in steps 202 and 206 allow the standard FDTD updatesfor diagonal media to be used.

Each finite element may optionally be represented by a Yee cell. The Yeelattice represented by FIG. 4 enables the electric field vectors to bespatially separated from the magnetic field vectors, which can simplifythe update performed by the update generation module 156 in block 204 ofFIG. 3. As is shown in FIG. 4, the Yee lattice comprises an electricfield element for each dimension and a separate magnetic field elementfor each dimension. Specifically, 302, 304, and 306 are electric fieldelements for the x, z, and y axis respectively, whereas 308, 310, and312, are magnetic field element for each of the x, z, and y axisrespectively.

A mesh comprising a Yee cell lattice enables the calculation of theelectric field updates and the magnetic field updates to be staggeredsuch that the curl equations can be solved in a numerically efficientway. In addition, the electric field (E) and magnetic field (H) updatescan be staggered in time to solve Maxwell's equations. Although a Yeecell is represented in FIG. 4, other cell representations may otherwisebe used.

Using Yee cells or other cells having separated magnetic field andelectric field components can reduce the computational power required tosimulate a medium using FDTD methods.

For example, when solving three-dimensional equations, each Yee cellcomprises three electric field elements and three magnetic fieldelements. Therefore, if there are two Yee cells in a simulation, boththe electric fields and magnetic fields in the Yee cells would bedescribed by six elements. Simulations might comprise hundreds orthousands or more of Yee cells. Both the electric field and magneticfield in a simulation comprise 3n elements, where n is the number of Yeecells.

In the FDTD method, the permittivity operator, or inverse permittivityoperator, relates the 3n components of the displacement field to the 3ncomponents of the electric field.

The general permittivity update, which may be applied by the updategeneration module 156, is shown below:

$\begin{pmatrix}D_{1,1,1}^{x} \\D_{1,1,1}^{y} \\D_{1,1,1}^{z} \\D_{2,1,1}^{x} \\D_{2,1,1}^{y} \\D_{2,1,1}^{z} \\\ldots \\D_{{nx},{ny},{nz}}^{x} \\D_{{nx},{ny},{nz}}^{y} \\D_{{nx},{ny},{nz}}^{z}\end{pmatrix} = {ɛ\begin{pmatrix}E_{1,1,1}^{x} \\E_{1,1,1}^{y} \\E_{1,1,1}^{z} \\E_{2,1,1}^{x} \\E_{2,1,1}^{y} \\E_{2,1,1}^{z} \\\ldots \\E_{{nx},{ny},{nz}}^{x} \\E_{{nx},{ny},{nz}}^{y} \\E_{{nx},{ny},{nz}}^{z}\end{pmatrix}}$

where i, j, and k represent the position of the corresponding Yee cellon the 3D FDTD mesh. When solving for isotropic or diagonal anisotropicmedia the permittivity or permeability tensor is diagonal when thereference frame is aligned with the axis of the anisotropic medium. Tosolve for the more general anisotropy, it may be computationallyefficient to use a block diagonal form of the permittivity tensor, asshown below:

$\begin{pmatrix}D_{1,1,1}^{x} \\D_{1,1,1}^{y} \\D_{1,1,1}^{z} \\D_{2,1,1}^{x} \\D_{2,1,1}^{y} \\D_{2,1,1}^{z} \\\ldots \\D_{{nx},{ny},{nz}}^{x} \\D_{{nx},{ny},{nz}}^{y} \\D_{{nx},{ny},{nz}}^{z}\end{pmatrix} = {\begin{pmatrix}\begin{pmatrix}ɛ_{xx} & ɛ_{xy} & ɛ_{xz} \\ɛ_{yx} & ɛ_{yy} & ɛ_{yz} \\ɛ_{zx} & ɛ_{zy} & ɛ_{zz}\end{pmatrix}_{1,1,1} & 0 & \ldots & 0 \\0 & \begin{pmatrix}ɛ_{xx} & ɛ_{xy} & ɛ_{xz} \\ɛ_{yz} & ɛ_{yy} & ɛ_{yz} \\ɛ_{zx} & ɛ_{zy} & ɛ_{zz}\end{pmatrix}_{2,1,1} & \ldots & 0 \\\ldots & \ldots & \ldots & \ldots \\0 & 0 & \ldots & \begin{pmatrix}ɛ_{xx} & ɛ_{xy} & ɛ_{xz} \\ɛ_{yz} & ɛ_{yy} & ɛ_{yz} \\ɛ_{zx} & ɛ_{zy} & ɛ_{zz}\end{pmatrix}_{{nx},{ny},{nz}}\end{pmatrix}\begin{pmatrix}E_{1,1,1}^{x} \\E_{1,1,1}^{y} \\E_{1,1,1}^{z} \\E_{2,1,1}^{x} \\E_{2,1,1}^{y} \\E_{2,1,1}^{z} \\\ldots \\E_{{nx},{ny},{nz}}^{x} \\E_{{nx},{ny},{nz}}^{y} \\E_{{nx},{ny},{nz}}^{z}\end{pmatrix}}$

Although the block diagonal form may simplify the field updates, aspatial asymmetry is introduced due to the location of the electric andmagnetic field vectors in each Yee cell.

As was mentioned above and illustrated in FIG. 4, each Yee cellcomprises three magnetic field elements and three electric fieldelements, none of which at exactly the same location. Iterativelyupdating the electric and magnetic fields according to the aboveequation introduces a bias from the resulting magnetic and electricfields in the Yee cell.

However, this bias can be overcome using several methods. For example,the update generation module 156 could employ various unitarytransformation interpolation approaches to transform neighbouring Yeecells, however, this approach would be likely to introduce non-diagonalterms in the permittivity in such a way that it would be challenging tofind a convenient block diagonal form, thereby complicating the solutionof the electric field and magnetic field updates described above.

Another approach to mitigate the bias is for the update generationmodule 156 to average several updates of the electric field and magneticfield alternating between the neighbouring cell on a first side of eachYee lattice and a neighbouring cell on a second side of each Yeelattice. For example, the update generation module 156 is operable toupdate the electric field using the average of several independentunitary transformations of the electric displacement field. For examplethe update generation module 156 is operable to employ followingrelationship to mitigate or eliminate the bias induced by using aparticular block diagonal form of the permittivity.

$\overset{arrow}{E} = {{\frac{1}{N}{\sum\limits_{i = 1}^{N}{\overset{arrow}{E}}_{i}}} = {{\frac{1}{N}U_{i}{\overset{arrow}{E}}_{i}^{\prime}} = {{\frac{1}{N}{\sum\limits_{i = 1}^{N}{{U_{i}( ɛ^{\prime} )}^{- 1}{\overset{arrow}{D}}_{i}^{\prime}}}} = {( {\frac{1}{N}{\sum\limits_{i = 1}^{N}{{U_{i}( ɛ^{\prime} )}^{- 1}U_{i}^{\dagger}}}} )\overset{arrow}{D}}}}}$

Taking the mean of the inverse permittivity term according to:

$( {\frac{1}{N}{\sum\limits_{i = 1}^{N}{{U_{i}( ɛ^{\prime} )}^{- 1}U_{i}^{\dagger}}}} )$

retains the properties of the individual permittivity tensors.Specifically, the average permittivity tensor will be symmetric,positive semidefinite, definite, or Hermitian as each inversepermittivity vector. By the update generation module 156 averaging overindividual permittivity tensors, the spatial bias can be eliminatedentirely to preserve the spatial symmetry of the magnetic and electricfields. However, the computational requirements are minimized since eachindividual permittivity can be written in block diagonal form.

Although the above is described with reference to linear materials, aperson skilled in the art would recognize that the same concepts couldbe extended to simulations of nonlinear materials. Despite the stabilityconditions of a nonlinear FDTD simulation potentially being moredifficult to satisfy, the stability conditions would not be made morecomplicated by performing the unitary coordinate transformation.Transformation of the coordinate system will not, in other words,negatively impact the stability of a simulation of a nonlinear medium.

Furthermore, although the above was described in the context of solvingMaxwell's equations using the FDTD method, the same approach can be usedto obtain other time domain and frequency domain solutions of Maxwell'sequations. For example, the method as described herein can be applied tosystems wherein all the electromagnetic field components are known atthe same spatial location (i.e. spatial elements were not represented bya Yee cell or other spatially separated representation). This cansimplify the application of local coordinate transformations by removingthe need to position the field components on a Yee cell or other similarcell typically used in FDTD simulations.

In most examples, it is advantageous to diagonalize the permittivity andpermeability tensors to solve Maxwell's equations, as this cansignificantly reduce the number of computation steps required and canfurther obviate the need to use grid smoothing techniques. The unitarytransformation method described herein also simplifies thetransformation between coordinate reference frames. This enables thetransformation module 150 to transform the local coordinate system to areference frame that simplifies or otherwise facilitates an updateprovided by the update generation module 156 to solve Maxwell'sequations.

The transformation module 150 may optionally be operable to transformthe coordinate system to yet another reference frame to performadditional calculations. For example, the transformation module 150 maytransform the coordinate system to solve Ampere's law and Faraday's lawin a reference frame consistent with a Cartesian mesh. Using thetransformation method described above, the local coordinate system canbe aligned with the Cartesian mesh without altering important propertiesof the tensors.

Generally, the unitary transformation can be used to transform thecoordinate system into other advantageous reference frames. For example,in a simulation having a first medium in contact with a second medium,it may be difficult for an update generation module to solve Maxwell'sequations along the interface. The transformation module 150 can apply aunitary transformation to the reference frame to align one of the axesof the coordinate system with the normal of the surface of theinterface. The remaining two axes of a three dimensional coordinatesystem will therefore lie substantially parallel to the surface of theinterface, simplifying the generation of updates to Maxwell's equationsby an update generation module.

Although coordinate system transformations have been described withreference to specific examples, it will be understood that othercoordinate system transformations are within the scope of thedescription. For example, the transformation module 150 may transformany coordinate system transformation to an updated reference frame tosimplify or otherwise facilitate a computation provided by the updategeneration module 156.

The ability to perform local transformations enables distincttransformations to be applied to various areas of the medium. Asmentioned above, this is particularly advantageous for LCD's where eachelement is aligned with the axis of the molecular axis of the liquidcrystal at a particular location. The same material models can beapplied to each mesh element, even if the orientation of individualliquid crystal molecules differs prior to the transformation. Becausemany LCD simulations involve millions of grid cells, this reduces thecomputational steps required to simulate these devices.

Time domain methods can be particularly challenging methods of solvingfor Maxwell's equations for dispersive media having non-diagonal termsin the permittivity tensor because the relationship between the electricdisplacement field and the electric field involve a convolution productin the time domain. Similarly, the relationship between the magneticflux density and the magnetic field involves a convolution product inthe time domain.

By way of example, {right arrow over (D)}(ω)=ε(ω){right arrow over(E)}(ω) in the frequency domain becomes the convolution product:

${\overset{arrow}{D}(t)} = {{{ɛ(t)}*{\overset{arrow}{E}(t)}} = {\int_{0}^{t}{{\overset{arrow}{E}( t^{\prime} )}{ɛ( {t - t^{\prime}} )}{t^{\prime}}}}}$

in the time domain, this is significantly more difficult to computenumerically. Specifically, for non-diagonal anisotropic media, there arethree times as many convolution products to solve.

However, a unitary transformation to change the reference frame anddiagonalize the permittivity minimizes the number of convolutionproducts to be solved by the update generation module 156. For example,a diagonalized matrix can have as little as a third of the number ofconvolution products that require a solution.

Importantly, for LCDs and other simulations where the diagonalpermittivity tensor is uniform across several elements (e.g. Yee cells),the same update algorithm and parameters can be used to evaluate theconvolution products across all elements upon the update generationmodule 156 solving for a single one of the plurality of identicalelements.

1-17. (canceled)
 18. A computer-implemented method for simulating anelectromagnetic property of a medium, the medium being meshed to definea plurality of mesh elements in an initial reference frame, the methodcomprising, for each mesh element: (a) determining a unitarytransformation at a transformation module having a processor configuredfor transforming an initial permittivity tensor in the initial referenceframe into a transformed permittivity tensor in a transformed referenceframe, the transformed permittivity tensor being diagonal; (b) applyingthe unitary transformation at the transformation module to an initialelectric displacement field tensor in the initial reference frame togenerate a transformed electric displacement field tensor in thetransformed reference frame; (c) communicating the transformedpermittivity tensor, the transformed electric displacement field tensorand a transformed electric field tensor to an update generation module;(d) generating an updated transformed electric field tensor at theupdate generation module using the transformed permittivity tensor, thetransformed electric displacement field tensor, and the transformedelectric field tensor; (e) communicating the updated transformedelectric field tensor to the transformation module; and (f) applying aninverse unitary transformation at the processor of the transformationmodule to the updated transformed electric field tensor to generate anupdated initial electric field tensor in the initial reference frame.19. The computer-implemented method according to claim 18, wherein step(b) further comprises applying the unitary transformation at thetransformation module to an initial electric field tensor in the initialreference frame to generate the transformed electric field tensor in thetransformed reference frame.
 20. The computer-implemented methodaccording to claim 18, wherein step (f) further comprises applying aninverse unitary transformation at the transformation module to thetransformed electric displacement field tensor.
 21. Thecomputer-implemented method according to claim 18, wherein the medium isanisotropic and the transformed reference frame is aligned with ananisotropic axis of the medium.
 22. The computer-implemented methodaccording to claim 18, wherein the initial reference frame is a globalcoordinate system.
 23. The computer-implemented method according toclaim 18, wherein the transformed reference frame is a local coordinatesystem.
 24. The computer-implemented method according to claim 18,wherein each mesh element is a Yee cell.
 25. A computer-implementedmethod for simulating an electromagnetic property of a medium, themedium being meshed to define a plurality of mesh elements in an initialreference frame, the method comprising, for each mesh element: (a)determining a unitary transformation at a transformation module having aprocessor configured for transforming an initial permeability tensor inthe initial reference frame into a transformed permeability tensor in atransformed reference frame, the transformed permeability tensor beingdiagonal; (b) applying the unitary transformation at the transformationmodule to an initial magnetic flux density tensor in the initialreference frame to generate a transformed magnetic flux density tensorin the transformed reference frame; (c) communicating the transformedpermeability tensor, the transformed magnetic flux density tensor and atransformed magnetic field tensor to an update generation module; (d)generating an updated transformed magnetic field tensor at the updategeneration module using the transformed permeability tensor, thetransformed magnetic flux density tensor, and the transformed magneticfield tensor; (e) communicating the updated transformed magnetic fieldtensor to the transformation module; and (f) applying an inverse unitarytransformation at the processor of the transformation module to theupdated transformed magnetic field tensor to generate an updated initialmagnetic field tensor in the initial reference frame.
 26. Thecomputer-implemented method according to claim 25, wherein step (b)further comprises applying the unitary transformation at thetransformation module to an initial magnetic field tensor in the initialreference frame to generate the transformed magnetic field tensor in thetransformed reference frame.
 27. The computer-implemented methodaccording to claim 25, wherein step (f) further comprises applying aninverse unitary transformation at the transformation module to thetransformed magnetic flux density tensor.
 28. The computer-implementedmethod according to claim 25, wherein the medium is anisotropic and thetransformed reference frame is aligned with an anisotropic axis of themedium.
 29. The computer-implemented method according to claim 25,wherein the initial reference frame is a global coordinate system. 30.The computer-implemented method according to claim 25, wherein thetransformed reference frame is a local coordinate system.
 31. Thecomputer-implemented method according to claim 25, wherein each meshelement is a Yee cell.
 32. A non-transitory computer readable mediumhaving stored thereon computer readable instructions for causing aprocessor to simulate an electromagnetic property of a medium, themedium being meshed to define a plurality of mesh elements in an initialreference frame, the computer readable medium comprising instructionsfor: (a) determining, for each mesh element, a unitary transformation ata transformation module for transforming an initial permittivity tensorin the initial reference frame into a transformed permittivity tensor ina transformed reference frame, the transformed permittivity tensor beingdiagonal; (b) applying, for each mesh element, the unitarytransformation at the transformation module to an initial electricdisplacement field tensor in the initial reference frame to generate atransformed electric displacement field tensor in the transformedreference frame; (c) communicating, for each mesh element, thetransformed permittivity tensor, the transformed electric displacementfield tensor and a transformed electric field tensor to an updategeneration module; (d) generating, for each mesh element, an updatedtransformed electric field tensor at the update generation module usingthe transformed permittivity tensor, the transformed electricdisplacement field tensor, and the transformed electric field tensor;(e) communicating, for each mesh element, the updated transformedelectric field tensor to the transformation module; and (f) applying,for each mesh element, an inverse unitary transformation at thetransformation module to the updated transformed electric field tensorto generate an updated initial electric field tensor in the initialreference frame.
 33. A non-transitory computer readable medium havingstored thereon computer readable instructions for causing a processor tosimulate an electromagnetic property of a medium, the medium beingmeshed to define a plurality of mesh elements in an initial referenceframe, the computer readable medium comprising instructions for: (a)determining, for each mesh element, a unitary transformation at atransformation module for transforming an initial permeability tensor inthe initial reference frame into a transformed permeability tensor in atransformed reference frame, the transformed permeability tensor beingdiagonal; (b) applying, for each mesh element, the unitarytransformation at the transformation module to an initial magnetic fluxdensity tensor in the initial reference frame to generate a transformedmagnetic flux density tensor in the transformed reference frame; (c)communicating, for each mesh element, the transformed permeabilitytensor, the transformed magnetic flux density tensor and a transformedmagnetic field tensor to an update generation module; (d) generating,for each mesh element, an updated transformed magnetic field tensor atthe update generation module using the transformed permeability tensor,the transformed magnetic flux density tensor, and the transformedmagnetic field tensor; (e) communicating, for each mesh element, theupdated transformed magnetic field tensor to the transformation module;and (f) applying, for each mesh element, an inverse unitarytransformation at the transformation module to the updated transformedmagnetic field tensor to generate an updated initial magnetic fieldtensor in the initial reference frame.